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    Periodic and constant solutions of matrix Riccati differential equations: n — 2. Proc. Roy. Sur 1’equation differentielle matricielle de type Riccati. Bull. Math. The qualitative study of second order linear equations originated in the classic paper . for a history of the Riccati transformation. Differentielle. (Q(t),’)’. VESSIOT, E.: “Sur quelques equations diffeYentielles ordinaires du second ordre .” Annales de (3) “Sur l’equation differentielle de Riccati du second ordre.

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    Skip to main content. Log In Sign Up. Liouvillian propagators, Riccati equation and differential Galois theory. In this paper a Galoisian approach to build propagators through Riccati equations is presented. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.

    The fact that in quan- tum electrodynamics the electromagnetic field can be represented as a set of forced harmonic oscillators makes quadratic Hamiltonians of special interest [7, 8, 11, 14, 35, 39]. Cauchy problem with this approach have been studied in [21, 26]. Primary 81Q05; Secondary 12H Key words and phrases.

    Differential Galois group, Green functions, propagators, Ric- cati equation. This Galoisian structure depends on the nature of the solutions of the differential equation; for instance, one obtains some kind of solvability virtual for the Galois group whenever one obtains Liouvillian solutions, and in this case one says that the differential equation is inte- grable.

    This means for example that when one obtains Airy functions, the differential equation is not integrable, while when one obtains Jacobi elliptic functions, the differential equation is integrable, and for instance one gets virtual solvability of its Galois group. In this paper we present a Galoisian approach of how to find explicit propagators through Liouvillian solutions for linear second order differential equations associated to Riccati equations.

    The main application of this Ga- loisian approach, for instance a main result of the paper, is the construction of the propagator for the so called degenerate parametric oscillator: The oscil- lator 1 – 2 might be introduced for the first time by Takahasi [37] in order to describe the process of degenerate parametric amplification in quantum optics see also [19, 20, 22, 23, 30, 31, 37].

    The Hamiltonian 2 had also been considered by Angelow and Trifonov [4, 5] in order to describe the light propagation in a nonlinear anisotropic waveguide. In fact, they present a non-periodic solution that allows us to write the propagator explicitly; the fact of the solution being non-periodic is fundamental. We believe this approach can be extended to the study of propagators of other generalized harmonic oscillators, but here we restrict ourselves to 1 – 2 and give some toy examples in Section 5.

    The aim of this paper is to establish a Galoisian approach to the tech- niques given by Suslov et. To study Liouvillian solutions for linear second order differential equations, as well the integrability of their associated Riccati equations, we use Kovacic algorithm see [17] and an algebrization procedure see [1, 3]. This paper is organized in the following way: It is devoted to a theoretical Galoisian approach to propagators starting with Ric- cati and second order differential equations.

    The Galois theory of differential equations, also called Differential Galois Theory and Picard-Vessiot The- ory, has been developed by Picard, Vessiot, Kolchin and currently by a lot of researchers, see [2, 3, 15, 16, 17, 25, 38]. Moreover, recent applications to mathematical physics can be found in [1, 3, 27, 28, 32]. We consider the differential Galois theory in the context of second order linear differential equations.


    The coefficient field for a differ- ential differentie,le is defined as the smallest differential field containing all the coefficients of the equation.

    We denote by G0 the connected component of the identity, thus, when G0 satisfies some property, we say that G virtually satisfies such property. Theorem 2 Lie-Kolchin Theorem. Then G0 is triangularizable.

    The solutions obtained throgh such towers are called Liouvillian. The special functions are not always Liouvillian, we can see that Airy equation has not Ricati solutions, while Bessel equation has Equatiin solutions for special values of the parameter, see [13, 29].

    Thus, we mean by inte- grable whenever the differential equation has Liouvillian solutions instead of special functions solutions.

    Consider now the following forms associated to any second order differential equation ode and Riccati equation: From Proposition 4, it is recovered the well known result in differential Galois theory see for example [38]: Only for cases 1, 2 and 3 we can solve the differential equation, qeuation for the case 4 the differential equation is not integrable. We use the following notations: Now, we summarize the cases one and two of Kovacic Algorithm that will be used in Section 4.

    Else, case 1 cannot hold. If Pn does not exist, then case 2 cannot hold.

    We can see that applying by hand Kovacic Algorithm it can be a lit- tle difficults, but thanks to has been implemented it in Maple command kovacicsols we can avoid such calculations. The problem with Maple is that the answers can have complicated expressions that should be trans- formed into more suitable and readable expressions.

    Riccati equation

    On the other hand, Kovacic Algorithm only works with rational coefficients, for instance, when the differential equation has not rational coefficients we cannot apply Ko- vacic Algorithm. We present a short summary of the Hamiltonian algebrization process that will be used in Section 4.

    This procedure is called Hamiltonian Algebrization, which is an isogaloisian transformation, i. Further details and proofs can be found in [1, 3]. Propagators and Green Functions. Furthermore, by Proposition 4, departing from the differential equation riccai we can arrive at the Riccati equation 15 through changes of variables.

    The following lemmas show how can we construct propagators based on explicit solutions in 15 and Further, the following asymptotics hold: Then the superposition principle allows us to solve the corresponding Cauchy ini- tial value problem: Galoisian Approach to Propagators In this section we apply the Picard-Vessiot theory in the context of propagators. In this way, we can give a Galoisian for- mulation for this kind of integrability.

    Theorem 9 Galoisian approach to LSE. In virtue of Theorem 9 we can construct many Liouvillian propagators through integrable second order differential equations over a dif- ferential field as well by algebraic solutions, over such differential field, of Riccati equations.

    This is the practical aim of this paper, which will be given in Section 4 and in Section 5. Kovacic Diffreentielle developed by J. Ko- vacic in [17] and Hamiltonian Algebrization developed by the first author in [1, 3].

    Here we follow the version of Kovacic Algorithm given equatioon [1, 3, 17]. In this section we find solutions to 4.

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    Afterward, we con- struct the corresponding propagator associated to this characteristic equa- tion. We begin analyzing case one.

    Thus, we have proven the following result. Toy Examples In this section we illustrate our Galoisian approach through some ele- mentary examples. We use the Galoisian approach to LSE, Theorem 9, to generalize examples introduced in [18] as well as to introduce toy examples.


    The starting point is the knowledge of the integrability, in the Picard-Vessiot sense, of the characteristic equation, or equivalently the existence of an al- gebraic solution over its differential field of its associated Riccati equation.

    Toy models inspired by integrable Riccati equations. Now, we study the integrability, in Galoisian sense, of this equation through Kovacic Algorithm. As in [32], we obtain three conditions for n to get virtual solv- ability of the differential Galois group. It follows the construction of the propagators related with these integrability conditions. In [1] there is a complete study of the Galoisian structure of this equation. Final Remarks This paper is an starting point to study the integrability of partial differ- ential equations in a more general sense through differential Galois theory.

    Although there are a plenty of papers concerning to explicit solutions and harmonic oscilla- tor see [12]we recall that differential Galois theory can provide the Liou- villian solutions of characteristic equations without previous knowledge of such equations. This is different to construct the explicit propagators know- ing apriory the solutions of the Riccatti or characteristic equation, which can open other possibilities to study propagator with special functions as characteristic equations, for example, Heun equation.

    The second author acknowledges being a recipient of Becas Iberoamericanas, jovenes profesores e invetigadores Santander Universidades during The second author greatly appreciates the support by University of Puerto Rico and Universitat de Barcelona during the academic visit to the latter in Spring where this project was born.

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    Suazo also was supported by the AMS-Simons Travel Grants, with support provided by the Simons Foundation to finish this project and the continuation of others. Morales-Ruiz for their useful comments, suggestions and great hospitality. Pantazi, On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory, Preprint arXiv: Angelow, Light propagation in nonlinear waveguide and classical two-dimensional oscillator, Physica A — [6] R. Suslov, Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields, Lett.

    Suslov, Models of damped oscillators in quantum mechanics, Journal of Physical Mathematics 1S 16 pages. Suslov, Quantum integrals of motion for vari- able quadratic Hamiltonians, Annals of Physics 10, —; see also Preprint arXiv: Suslov, Time reversal for modified oscillators, Theoret- ical and Mathematical Physics 3, —; see also Preprint arXiv: Ince, Ordinary differential equations, Dover, New York, Hannay, Angle variable holonomy in adiabatic excursion of an integrable Hamil- tonian, J.

    Gen 18 —30 [15] I. Kaplansky, An introduction to differential algebra, Hermann, Paris, Symbolic Computation, 23— Beijing, China 51 — [20] C. Lo, Coherent-state propagator of the generalized time-dependent parametric oscillator Europhys. Suslov, The Cauchy problem for a forced harmonic oscillatorPreprint arXiv: Yariv, Quantum fluctuations and noise in parametric processes: Orszag, Quantum OpticsSpringer, Heidelberg, Suslov, Dynamical invariants for variable quadratic Hamiltonians, Physica Scripta 81 5, 11 pp ; see also Preprint arXiv: Takahasi, Information theory of quantum-mechanical channels Advances in Com- munication Systems: Theory and Applications A.

    Pandey, Exact quantum theory of a time-dependent bound Hamiltonian systems Phys. A 48, — P. Remember me on this computer.

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